3.283 \(\int x^3 \left (-\sqrt{1-x}-\sqrt{1+x}\right ) \left (\sqrt{1-x}+\sqrt{1+x}\right ) \, dx\)

Optimal. Leaf size=38 \[ -\frac{x^4}{2}-\frac{2}{5} \left (1-x^2\right )^{5/2}+\frac{2}{3} \left (1-x^2\right )^{3/2} \]

[Out]

-x^4/2 + (2*(1 - x^2)^(3/2))/3 - (2*(1 - x^2)^(5/2))/5

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Rubi [A]  time = 0.6136, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ -\frac{x^4}{2}-\frac{2}{5} \left (1-x^2\right )^{5/2}+\frac{2}{3} \left (1-x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-x^4/2 + (2*(1 - x^2)^(3/2))/3 - (2*(1 - x^2)^(5/2))/5

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(-(1-x)**(1/2)-(1+x)**(1/2))*((1-x)**(1/2)+(1+x)**(1/2)),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0443343, size = 44, normalized size = 1.16 \[ -\frac{1}{30} \left (x^2-1\right ) \left (3 \left (4 \sqrt{1-x^2}+5\right ) x^2+8 \sqrt{1-x^2}+15\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(-Sqrt[1 - x] - Sqrt[1 + x])*(Sqrt[1 - x] + Sqrt[1 + x]),x]

[Out]

-((-1 + x^2)*(15 + 8*Sqrt[1 - x^2] + 3*x^2*(5 + 4*Sqrt[1 - x^2])))/30

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Maple [A]  time = 0.003, size = 33, normalized size = 0.9 \[ -{\frac{{x}^{4}}{2}}-{\frac{ \left ( 2\,{x}^{2}-2 \right ) \left ( 3\,{x}^{2}+2 \right ) }{15}\sqrt{1-x}\sqrt{1+x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(-(1-x)^(1/2)-(1+x)^(1/2))*((1-x)^(1/2)+(1+x)^(1/2)),x)

[Out]

-1/2*x^4-2/15*(1+x)^(1/2)*(1-x)^(1/2)*(x^2-1)*(3*x^2+2)

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Maxima [A]  time = 0.766081, size = 42, normalized size = 1.11 \[ -\frac{1}{2} \, x^{4} + \frac{2}{5} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} x^{2} + \frac{4}{15} \,{\left (-x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3*(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="maxima")

[Out]

-1/2*x^4 + 2/5*(-x^2 + 1)^(3/2)*x^2 + 4/15*(-x^2 + 1)^(3/2)

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Fricas [A]  time = 0.271582, size = 109, normalized size = 2.87 \[ -\frac{12 \, x^{10} - 85 \, x^{8} + 80 \, x^{6} + 5 \,{\left (9 \, x^{8} - 16 \, x^{6}\right )} \sqrt{x + 1} \sqrt{-x + 1}}{30 \,{\left (5 \, x^{4} - 20 \, x^{2} -{\left (x^{4} - 12 \, x^{2} + 16\right )} \sqrt{x + 1} \sqrt{-x + 1} + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3*(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="fricas")

[Out]

-1/30*(12*x^10 - 85*x^8 + 80*x^6 + 5*(9*x^8 - 16*x^6)*sqrt(x + 1)*sqrt(-x + 1))/
(5*x^4 - 20*x^2 - (x^4 - 12*x^2 + 16)*sqrt(x + 1)*sqrt(-x + 1) + 16)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(-(1-x)**(1/2)-(1+x)**(1/2))*((1-x)**(1/2)+(1+x)**(1/2)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287635, size = 76, normalized size = 2. \[ -\frac{1}{2} \,{\left (x + 1\right )}^{4} + 2 \,{\left (x + 1\right )}^{3} - \frac{2}{15} \,{\left ({\left (3 \,{\left (x + 1\right )}{\left (x - 3\right )} + 17\right )}{\left (x + 1\right )} - 10\right )}{\left (x + 1\right )}^{\frac{3}{2}} \sqrt{-x + 1} - 3 \,{\left (x + 1\right )}^{2} + 2 \, x + 2 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^3*(sqrt(x + 1) + sqrt(-x + 1))^2,x, algorithm="giac")

[Out]

-1/2*(x + 1)^4 + 2*(x + 1)^3 - 2/15*((3*(x + 1)*(x - 3) + 17)*(x + 1) - 10)*(x +
1)^(3/2)*sqrt(-x + 1) - 3*(x + 1)^2 + 2*x + 2